### Abstract

Let K be a field of characteristic p and let G be a finite group of order divisible by p. The regularity conjecture states that the Castelnuovo-Mumford regularity of the cohomology ring H* (G, K) is always equal to 0. We prove that if the regularity conjecture holds for a finite group H, then it holds for the wreath product H similar to Z/p. As a corollary, we prove the regularity conjecture for the symmetric groups Sigma(n). The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

Original language | English |
---|---|

Pages (from-to) | 273-284 |

Number of pages | 12 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 |

### Keywords

- cohomology of groups
- local cohomology
- Castelnuovo-Mumford regularity
- wreath product
- spctrum
- ring

### Cite this

**On the regularity conjecture for the cohomology of finite groups.** / Benson, David J.

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 51, no. 2, pp. 273-284. https://doi.org/10.1017/S0013091505001203

}

TY - JOUR

T1 - On the regularity conjecture for the cohomology of finite groups

AU - Benson, David J.

PY - 2008

Y1 - 2008

N2 - Let K be a field of characteristic p and let G be a finite group of order divisible by p. The regularity conjecture states that the Castelnuovo-Mumford regularity of the cohomology ring H* (G, K) is always equal to 0. We prove that if the regularity conjecture holds for a finite group H, then it holds for the wreath product H similar to Z/p. As a corollary, we prove the regularity conjecture for the symmetric groups Sigma(n). The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

AB - Let K be a field of characteristic p and let G be a finite group of order divisible by p. The regularity conjecture states that the Castelnuovo-Mumford regularity of the cohomology ring H* (G, K) is always equal to 0. We prove that if the regularity conjecture holds for a finite group H, then it holds for the wreath product H similar to Z/p. As a corollary, we prove the regularity conjecture for the symmetric groups Sigma(n). The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

KW - cohomology of groups

KW - local cohomology

KW - Castelnuovo-Mumford regularity

KW - wreath product

KW - spctrum

KW - ring

U2 - 10.1017/S0013091505001203

DO - 10.1017/S0013091505001203

M3 - Article

VL - 51

SP - 273

EP - 284

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -